P7F.8 (a) Show that y=c₁Y+c is an eigenfunction of A² with eigenvalue -1(1+1); c, and care arbitrary coefficients. (Hint: Apply A² to and use the properties given in eqn 7F.9.) (b) The spherical harmonics Y., and Y., are complex functions (see Table 7F.1), but as they are degenerate eigenfunctions of A², any linear combination of them is also an eigenfunction, as was shown in (a). Show that the combinations w₁ =-Y+Y and wi(Y₁. + Y₁-1) are real. (c) Show that w, and y, are orthogonal (you will need to integrate using the relevant volume element, see Problem P7F.7). (d) Normalize w and y. (e) Identify the angular nodes in these two functions and the planes to which they correspond. (f) Is w, an eigenfunction of I? Discuss the significance of your answer.